A237979 - OEIS

A237979

Number of strict partitions of n such that (least part) > number of parts.

0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 7, 7, 9, 10, 12, 13, 16, 17, 20, 22, 25, 28, 32, 35, 40, 45, 50, 56, 63, 70, 78, 87, 96, 107, 118, 131, 144, 160, 175, 194, 213, 235, 257, 284, 310, 342, 373, 410, 447, 491, 534, 585, 637, 696, 756, 826, 896, 977, 1060, 1153, 1250, 1359, 1471, 1597, 1729, 1874, 2026, 2195, 2371, 2565

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OFFSET

1,7

COMMENTS

Also the number of partitions into distinct parts with minimal part >= 2 and difference between parts >= 3. [Joerg Arndt, Mar 31 2014]

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000

FORMULA

G.f. with a(0)=0: sum(n>=0, q^(n*(3*n+1)/2) / prod(k=1..n, 1-q^k ) ). [Joerg Arndt, Mar 09 2014]

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(Pi*(1 + 3*r^2)) * n^(3/4)), where r = A263719 and c = 3*(log(r))^2/2 + polylog(2, 1-r). - Vaclav Kotesovec, Jan 15 2022

a(n) ~ A263719 * A025157(n). - Vaclav Kotesovec, Jan 15 2022

EXAMPLE

a(9) = 3 counts these partitions: 9, 63, 54.

MATHEMATICA

z = 50; q[n_] := q[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];

p1[p_] := p1[p] = DeleteDuplicates[p]; t[p_] := t[p] = Length[p1[p]]

Table[Count[q[n], p_ /; Min[p] < t[p]], {n, z}] (* A237976 *)

Table[Count[q[n], p_ /; Min[p] <= t[p]], {n, z}] (* A237977 *)

Table[Count[q[n], p_ /; Min[p] == t[p]], {n, z}] (* A096401 *)

Table[Count[q[n], p_ /; Min[p] > t[p]], {n, z}] (* A237979 *)

Table[Count[q[n], p_ /; Min[p] >= t[p]], {n, z}] (* A025157 *)

PROG